I am very familiar with the concepts of bijective, surjective and injective maps but I am interested in improvising the definition of bijection in a way I have not seen done before. To be clear I will provide the definitions that I use for surjective, injective and bijective although they are the common and generally accepted definitions.
$f:A to B$ is $surjective$ $iff$ $forall y in B$, $exists x in A$ : $f(x) = y$.
$f:A to B$ is $injective$ $iff$ $forall x, y in A,$ $ f(x)=f(y) implies x=y$
$f:A to B$ is $bijective$ $iff$ ($forall y in B$, $exists x in A$ : $f(x) = y$) $wedge$ ($forall x, y in A,$ $f(x)=f(y) implies x=y$)
You can see that the definition provided for bijection is simply the conjunction of surjective and injection.
My question is, does it suffice to define bijection as such:
$f:A to B$ is $bijective$ $iff$ $forall y in B$, $exists! x in A$ : $f(x) = y$.
Sorry for making a mountain out of a mole hill with this question, the person I usually work with is not here tonight so I have to run my spit ball ideas by you guys! Thanks in advance.